Form estimates for the $p(x)$-Laplacean

We consider the problem of establishing conditions on p(x) that ensure that the form associated with the p(x)-Laplacean is positive bounded below. It was shown recently by Fan, Zhang and Zhao that - unlike the p = constant case - this is not possible if p has a strict extrema in the domain. They also considered the closely related problem of eigenvalue existence and estimates. Our main tool is the adaptation of a technique, employed by Protter for p = 2, involving arbitrary vector fields. We also examine related results obtained by a variant of Picone Identity arguments. We directly consider problems in Ω C R n with n > 1, and while we focus on Dirichlet boundary conditions we also indicate how our approach can be used in cases of mixed boundary conditions, of unbounded domains and of discontinuous p(x). Our basic criteria involve restrictions on p(x) and its gradient.